Spring 2010 CS 225 1

Sorting

Chapter 8

Spring 2010 CS 225 2

Chapter Objectives• To learn how to use the standard sorting

methods in the Java API• To learn how to implement the following

sorting algorithms: selection sort, bubble sort, insertion sort, Shell sort, merge sort, heapsort, and quicksort

• To understand the difference in performance of these algorithms, and which to use for small arrays, which to use for medium arrays, and which to use for large arrays

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Declaring a Generic Method• We have seen generic classes throughout the

semester.• You can also declare generic methods.

methodModifiers <generic parameters> returnType methodName( methodParameters)

• List the generic type(s) between the modifiers and the return type; use them in the parameter list, return type(?) and body of method as needed.public static <T extends Comparable<T>> int binarySearch( T[] items, T target)

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Using Java Sorting Methods

• Java API provides a class Arrays with several overloaded sort methods for different array types

• The Collections class provides similar sorting methods for Lists

• Sorting methods for arrays of primitive types are based on quicksort algorithm

• Method of sorting for arrays of objects and Lists based on mergesort

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java.util.Arrays

• Two versions of sort for each primitive typepublic void sort( int [] values)public void sort( int [] values, int from, int to)

• Four sort methods for objectspublic static void sort( Object [] items)public static void sort( Object [] items, int from, int to)public static <T> void sort( <T> [] items, Comparator<? super T> comp)public static <T> void sort( <T> [] items, Comparator<? super T> comp, int from, int to)

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java.util.Collections

• Two methods for sorting collections that implement the List interfacepublic static

<T extends Comparable<T>>

void sort( List<T>list)

public static <T> void sort(

List<T> list,

Comparator<? super T> comp)

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Selection Sort

• Selection sort is a relatively easy to understand algorithm

• Sorts an array by making several passes through the array, selecting the next smallest item in the array each time and placing it where it belongs in the array

• Efficiency is O(n*n)

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Selection Sort Algorithm

• for fill = 0 to n-2 do

• posMin = fill

• for next = fill + 1 to n-1

• if item at next < item at posMin

• posMin = next

• swap elements at fill and posMin

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Selection Sort Algorithm

• Basic rule: on each pass select the smallest remaining item and place it in its proper location

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Selection Sort Efficiency

• Selection sort is called a quadratic sort– Number of comparisons is O(n*n)– Number of exchanges is O(n)

• In best case (already sorted)– Number of comparisons is still O(n*n)– Number of exchanges is O(1)

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Bubble Sort

• Compares adjacent array elements and exchanges their values if they are out of order

• Smaller values bubble up to the top of the array and larger values sink to the bottom

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Analysis of Bubble Sort• Provides excellent performance in some

cases and very poor performances in other cases

• Works best when array is nearly sorted to begin with

• Worst case number of comparisons is O(n*n)• Worst case number of exchanges is O(n*n)• Best case occurs when the array is already

sorted– O(n) comparisons– O(1) exchanges

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Insertion Sort

• Based on the technique used by card players to arrange a hand of cards– Player keeps the cards that have been

picked up so far in sorted order– When the player picks up a new card, he

makes room for the new card and then inserts it in its proper place

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Insertion Sort Algorithm

• For each array element from the second to the last (nextPos = 1)– Insert the element at nextPos where it

belongs in the array, increasing the length of the sorted subarray by 1

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Analysis of Insertion Sort• Maximum number of comparisons is O(n*n)• In the best case (already sorted), number of

comparisons is O(n)• The number of shifts performed during an insertion is

one less than the number of comparisons or, when the new value is the smallest so far, the same as the number of comparisons

• A shift in an insertion sort requires the movement of only one item whereas in a bubble or selection sort an exchange involves a temporary item and requires the movement of three items

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Comparison of Quadratic Sorts

• None of the algorithms are particularly good for large arrays

Comparisons Exchanges

Sort Best Worst Best Worst

Selection O(n2) O(n2) O(n) O(n)

Bubble O(n) O(n2) O(1) O(n2)

Insertion O(n) O(n2) O(n) O(n2)

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Shell Sort

• Shell sort is a type of insertion sort but with O(n(3/2)) or better performance

• Named after its discoverer, Donald Shell

• Divide and conquer approach to insertion sort– Instead of sorting the entire array, sort

many smaller subarrays using insertion sort before sorting the entire array

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Shell Sort

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Analysis of Shell Sort• A general analysis of Shell sort is an open

research problem in computer science• Performance depends on how the decreasing

sequence of values for gap is chosen• If successive powers of two are used for gap,

performance is O(n*n)• If Hibbard’s sequence is used, performance is

O(n3/2)

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Merge Sort

• A merge is a common data processing operation that is performed on two sequences of data with the following characteristics– Both sequences contain items with a

common compareTo method– The objects in both sequences are ordered

in accordance with this compareTo method

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Merge Algorithm• Merge Algorithm

– Access the first item from both sequences– While not finished with either sequence

• Compare the current items from the two sequences, copy the smaller current item to the output sequence, and access the next item from the input sequence whose item was copied

– Copy any remaining items from the first sequence to the output sequence

– Copy any remaining items from the second sequence to the output sequence

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Merge Example

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Analysis of Merge

• For two input sequences that contain a total of n elements, we need to move each element’s input sequence to its output sequence– Merge time is O(n)

• We need to be able to store both initial sequences and the output sequence– The array cannot be merged in place– Additional space usage is O(n)

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Algorithm and Trace of Merge Sort

• if tablesize >1

• halfsize = tablesize / 2

• copy left half to separate leftTable

• recursively sort leftTable

• copy right half to separate rightTable

• recursively sort rightTable

• merge leftTable and rightTable

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Trace of Merge Sort

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Heapsort

• Merge sort time is O(n log n) but still requires, temporarily, n extra storage items

• Heapsort does not require any additional storage– build a max-heap from the array– swap first element with last to put largest

element at end– build a heap from the unsorted part of the

array

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Heapsort

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Heapsort

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Quicksort

• Developed in 1962• Quicksort rearranges an array into two parts

so that all the elements in the left subarray are less than or equal to a specified value, called the pivot

• Quicksort ensures that the elements in the right subarray are larger than the pivot

• Average case for Quicksort is O(n log n)

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Quicksort

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Algorithm for Partitioning

• pivot = table[first]• up = first• down = last• while table[up]<=pivot and up<down• increment up• while table[down]>pivot and down>up• decrement down• swap table[down] and table[first]• pivotIndex = down

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Algorithm for Partitioning

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Revised Partition Algorithm

• Quicksort is O(n*n) when each split yields one empty subarray, which is the case when the array is presorted

• Best solution is to pick the pivot value in a way that is less likely to lead to a bad split– Requires three markers

• First, middle, last

– Select the median of the these items as the pivot

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Testing the Sort Algorithms

• Need to use a variety of test cases– Small and large arrays– Arrays in random order– Arrays that are already sorted– Arrays with duplicate values

• Compare performance on each type of array

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Comparing Sorts

Sort Best Average Worst

Selection O(n2) O(n2) O(n2)

Bubble O(n) O(n2) O(n2)

Insertion O(n) O(n2) O(n2)

Shell O(n7/6) O(n5/4) O(n2)

Merge O(n log n) O(n log n) O(n log n)

Heap O(n log n) O(n log n) O(n log n)

Quick O(n log n) O(n log n) O(n2)

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The Dutch National Flag Problem

• A variety of partitioning algorithms for quicksort have been published

• A partitioning algorithm for partitioning an array into three segments was introduced by Edsger W. Dijkstra

• Problem is to partition a disordered three-color flag into the appropriate three segments

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The Dutch National Flag Problem

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Algorithm• for 0<=i<red, color is

red• for red<=i<=white, color

is unknown• for white<i<=blue, color

is blue• for blue<i<=height, color

is blue

• red = 0• blue = white = height-1• while red<white• if threads[white] is• white: white--• red: • swap elements at • red and white• red++ • blue:• swap elements at • blue and white• blue--• white--

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Chapter Review• Comparison of several sorting algorithms

were made• Three quadratic sorting algorithms are

selection sort, bubble sort, and insertion sort• Shell sort gives satisfactory performance for

arrays up to 5000 elements• Quicksort has an average-case performance

of O(n log n), but if the pivot is picked poorly, the worst case performance is O(n*n)

• Merge sort and heapsort have O(n log n) performance

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Chapter Review

• The Java API contains “industrial strength” sort algorithms in the classes java.util.Arrays and java.util.Collections